Question

In the diagram, segment AD bisects angle BAC.

Given the following segment lengths,
find the value of x.
Round to the nearest tenth.
AB= 23
AC = 18
Show all your work.

Answer 1

Answer: x ≅ 11.2

Explanation:

We can set up two equations:

Let y = measure of <BAD = measure of <CAD

then:

sin y = x/23

sin y = (20-x)/18

Since both of these are sin y, we can set them equal to each other:

x/23 = (20-x)/18

.: 18x = 460 - 23x

41x = 460

.: x ≅ 11.2

Answer 2

x = 11.2

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Use angle bisector theorem. It states that:

• An angle bisector of an angle of a triangle divides the opposite side into two parts that are proportional to the other two sides of the triangle.

Apply this to the given triangle:

• AB/AC = BD/CD
• 23/18 = x / (20 - x)

Cross-multiply and solve for x:

• 23(20 - x) = 18x
• 460 - 23x = 18x
• 460 = 41x
• x = 460/41
• x = 11.2 (rounded)